At the end of the year $2000$ the population of a city was $3000000$. The population then increased by $1.3\%$ per year. Find the population at the end of $2001$and $2002$ . Write an exponential function for $P\left(t\right)$, the population after $t$ years. Predict the year in which the population should exceed $350000$ .

A group of $15$ snow foxes are introduced into a nature reserve. The number of snow foxes $N$ can be modelled by the exponential equation $N=15\times 3^{0.4t}$ where $t$ is the number of years since their introduction. 

Find how many years after their introduction there should be at least $100$ snow foxes.

The half-life of caffeine in the human body is approximately $5$ hours.

Copy and complete the table for a student who consumes $120\mathrm{mg}$ of caffeine at $08: 00$ one morning.

$\begin{array}{|ccc|}\hline \text{ Time }& \begin{array}{c}\text{ Time period }\\ \text{ (in 5-hour intervals) }\\ \text{ (t hours) }\end{array}& \begin{array}{c}\text{ Amount of }\\ \text{ caffeine (C }\mathbf{mg})\end{array}\\ 08:00& t=0& 120\\ 13:00& t=1& 60\\ 18:00& t=2& \\ 23:00& & \\ \hline\end{array}$

The half-life of caffeine in the human body is approximately $5$ hours.

Copy and complete the table for a student who consumes $120\mathrm{mg}$ of caffeine at $08: 00$ one morning.

$\begin{array}{|ccc|}\hline \text{ Time }& \begin{array}{c}\text{ Time period }\\ \text{ (in 5-hour intervals) }\\ \text{ (t hours) }\end{array}& \begin{array}{c}\text{ Amount of }\\ \text{ caffeine (C }\mathbf{mg})\end{array}\\ 08:00& t=0& 120\\ 13:00& t=1& 60\\ 18:00& t=2& \\ 23:00& & \\ \hline\end{array}$

Find the common ratio for the geometric sequence.

The half-life of caffeine in the human body is approximately $5$ hours.

Copy and complete the table for a student who consumes $120\mathrm{mg}$ of caffeine at $08: 00$ one morning.

$\begin{array}{|ccc|}\hline \text{ Time }& \begin{array}{c}\text{ Time period }\\ \text{ (in 5-hour intervals) }\\ \text{ (t hours) }\end{array}& \begin{array}{c}\text{ Amount of }\\ \text{ caffeine (C }\mathbf{mg})\end{array}\\ 08:00& t=0& 120\\ 13:00& t=1& 60\\ 18:00& t=2& \\ 23:00& & \\ \hline\end{array}$

Find the common ratio for the geometric sequence.

Find the exponential equation for the half-life of caffeine in terms of $t$

The half-life of caffeine in the human body is approximately $5$ hours.

Copy and complete the table for a student who consumes $120\mathrm{mg}$ of caffeine at $08: 00$ one morning.

$\begin{array}{|ccc|}\hline \text{ Time }& \begin{array}{c}\text{ Time period }\\ \text{ (in 5-hour intervals) }\\ \text{ (t hours) }\end{array}& \begin{array}{c}\text{ Amount of }\\ \text{ caffeine (C }\mathbf{mg})\end{array}\\ 08:00& t=0& 120\\ 13:00& t=1& 60\\ 18:00& t=2& \\ 23:00& & \\ \hline\end{array}$

Find the common ratio for the geometric sequence.

Find the exponential equation for the half-life of caffeine in terms of $t$

Calculate the amount of caffeine in the student's body after $35$ hours

Coco is training for a race and records her $10 \mathrm{km}$ times each week. Before she starts training, she can run $10 \mathrm{km}$ in $72$ minutes.

$\begin{array}{|cc|}\hline \text{ Week }\left(w\right)& \text{ Time (t, min) to run }\mathbf{10}\mathbf{km}\\ 1& 70.560\\ 2& 69.149\\ 3& 67.766\\ 4& 66.411\\ 5& 65.082\\ \hline\end{array}$

Assuming her time continues to decrease at the same rate, calculate how many weeks it will be before Coco can run $10 \mathrm{km}$ in less than one hour.

A large lake has a population of $100000$ perch. The number of perch is decreasing at a rate of $10\%$ per year due to pollution.
Make a table to show the number of perch $P$ after $1,2$ and $3$ years.

Write an exponential model for the number of perch $P$ after $t$ years.